Is the the Hindu Trimurti (Brahma - Vishnu - Mahesh) the same as the the Christian doctrine of the Holy Trinity?
Are these 3 sides of one god?
While the Hindu concept of the trimūrti may superficially resemble the Christian Trinity in that both involve three members of a single Godhead, the concepts are far more different than similar. The individual members of the Hindu trimūrti are just illusory ways of conceiving the one same divine spirit of Brahman, and one can say that they are three sides to the same God, whereas the Christian Trinity involves three substantially different persons from the same Godhead that are one in will, action, knowledge (for the most part), and essence--the three persons of the Christian Trinity are not three aspects or faces of the same God.
The concept of the trimūrti (which does not belong to all Hindu ways of thought) is not exactly like the Christian concept of the Trinity. While in Hindu thought, the three members of the trimūrti are three sides to one God or Spirit, the three persons of the Christian Trinity are NOT three "aspects" or "sides" to one God--they are substantially distinct.
1) The Christian Trinity has three different persons, with real differences between each person. However, the Hindu conception of the trimūrti is similar to the Christian heresy of Sabellianism or modalism—that is, Viṣṇu, Brahmā, and Śiva are different aspects of Brahman, rather than being distinct persons, or different ways of conceiving the same Brahman, who acts in different roles at different times.
In the Christian Trinity, God the Son is begotten of God the Father, but not made from him; the Holy Ghost proceeds from the Father through the Son, so to speak. They are different persons of the same God, with substantial differences—the Threeness of God is not some kind of illusion in Christianity.
In the trimūrti, Viṣṇu, Brahmā, and Śiva are three sides to the same “Spirit" (Brahman) but they are just different conceptions/aspects/faces/modes/parts. Swami Vishwananda, who commented above on your question, said it perfectly: "The Hindu trimūrti (saguṇa brahman) is the nirguṇa brahman when seen from the aspect of māyā. In reality, there is no difference." However, the Christian Trinity involves the supreme harmony and cooperation of three distinct persons of the same God, who have different familial relationships.
2) Viṣṇu, Brahmā, and Śiva, when acting as members of the trimūrti, do not always cooperate, according to some parts of Hindu scripture. However, the three persons of the Christian God always cooperate and share the same attributes—omniscience, omnipotence, et cetera--although God the Son may have had some qualities limited, or voluntarily limited some qualities, like his omniscience and his omnipresence, subsequent to the hypostatic union (that is, the union of the divine nature and will of God the Son with his human nature and will).
3) The roles of the members of each triad do not correspond. Viṣṇu is Brahman in the role of preservation, Brahmā is Brahman in the role of creation/generation, and Śiva is Brahman in the role of (impermanent) destruction. In Christianity, God the Father, God the Son, and the Holy Spirit ALL are involved in any action by God. Their wills are tied together (with God the Son, his divine will is tied together with the will of God; his human will is separate and unrelated) always, and so is their knowledge (except in some cases where God the Son chose to limit his omniscience--for example, regarding the timing of his Second Coming).